Find the position satisfying a condition.

Geometry Level pending

An equilateral triangle $ABC$ is inscribed in a circle where $O$ is the center. Draw a random point $D$ on the minor arc $\stackrel \frown {BC}$ and a point $K$ on $DA$ such that $DK=DB$ .

Where on the circumference of the circle should you place the point $D$ so that the total length of $AD, BD, CD$ achieves the maximum value?

The total length is always constant, so the position of

D

doesn't matter Place

D

at any place between

A

and

C

Place

D

on the point

B

. Place

D

such that

AD

is the diameter of the circle

1 solution

Tin Le
Sep 7, 2020

The two inscribed angles $\widehat{BDA}, \widehat{BCA}$ share the same endpoints in the circle, so $\widehat{BDA}=\widehat{BCA}$

As $ABC$ is an equilateral triangle, $\widehat{BCA}=60^o=\widehat{BDA}$

Examining the isosceles triangle $BDK$ (As $BD = DK$ ), it has a $60^o$ angle. Hence, $BDK$ is an equilateral triangle.

Therefore, $BD = DK$ and $\widehat{DBK}=60^o$

We can see that $\widehat{ABC}=\widehat{KBD} (= 60^o) \Leftrightarrow \widehat{ABK} +\widehat{KBC}=\widehat{KBC}+\widehat{CBD} \Leftrightarrow \widehat{ABK}=\widehat{CBD}$

Examining the two triangles $ABK$ and $CBD$ :

$AB=BC$ ( $BDK$ is an equilateral triangle)
$BD = DK$
$\widehat{ABK}=\widehat{CBD}$

Hence the two triangles are equal. (side-angle-side) $\Rightarrow AK = DC$

We have $AD = AK + KD = CD + BD$ .

Therefore $AD+BD+CD = 2AD$ .

We know that $AD$ is a chord. Therefore, the only scenario where $2AD$ achieves the maximum value is when $\boxed{\text{AD is the diameter}}$ .

Find the position satisfying a condition.

1 solution

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